Modeling workflow for symmetrical indented plane
starting with two objects like you describe:

Add a shrinkwrap modifier

to get something like this:

Some faces look beveled because the quads forming the mesh cannot accommodate the deformation properly.
To correct the topology of the edges in question, there are two options:

1: Press Ctrl T to triangulate the mesh.



2: Manually subdivide the offending faces selecting vertices and using J to join the edges.